Answer:
[tex]9^{-7}[/tex] and [tex]9^{-2}[/tex]
Step-by-step explanation:
using the rules of exponents
• [tex](a^m)^{n}[/tex] = [tex]a^{mn}[/tex]
• [tex]a^{m}[/tex] × [tex]a^{n}[/tex] = [tex]a^{(m+n)}[/tex]
• [tex]\frac{a^{m} }{a^{n} }[/tex] = [tex]a^{(m-n)}[/tex]
given ([tex]9^{5}[/tex] × [tex]9^{-9}[/tex] ) / [tex]9^{3}[/tex]
= [tex]\frac{9^{5+(-9)} }{9^{3} }[/tex] = [tex]\frac{9^{-4} }{9^{3} }[/tex] = [tex]9^{-4-3}[/tex] = [tex]9^{-7}[/tex]
given [tex](9^4)^{3}[/tex] × [tex]9^{-14}[/tex]
= [tex]9^{12}[/tex] × [tex]9^{-14}[/tex]
= [tex]9^{12+(-14)}[/tex] = [tex]9^{-2}[/tex]
Answer: The required values of the given expressions are
[tex] \dfrac{9^5.9^{-9}}{9^3}=9^{-7},~~~(9^4)^3.9^{-14}=9^{-2}.[/tex]
Step-by-step explanation: We are given to find the values of the following expressions :
[tex]E_1=\dfrac{9^5.9^{-9}}{9^3}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\\\E_2=(9^4)^3.9^{-14}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]
We will be using the following properties of exponents :
[tex](i)~x^a.x^b=x^{a+b},\\\\(ii)~\dfrac{x^a}{x^b}=x^{a-b}\\\\(iii)~(x^a)^b=x^{ab}.[/tex]
From expression (i), we get
[tex]E_1\\\\\\=\dfrac{9^5.9^{-9}}{9^3}\\\\\\=\dfrac{9^{5+(-9)}}{9^3}\\\\\\=\dfrac{9^{-4}}{9^3}\\\\\\=9^{-4-3}\\\\=9^{-7}[/tex]
and from expression (ii), we get
[tex](9^4)^3.9^{-14}\\\\=9^{4\times3}.9^{-14}\\\\=9^{12}.9^{-14}\\\\=9^{12+(-14)}\\\\=9^{-2}.[/tex]
Thus, the required values of the given expressions are
[tex] \dfrac{9^5.9^{-9}}{9^3}=9^{-7},~~~(9^4)^3.9^{-14}=9^{-2}.[/tex]
